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Theorem recexprlemex 7635
Description: 𝐵 is the reciprocal of 𝐴. Lemma for recexpr 7636. (Contributed by Jim Kingdon, 27-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
Assertion
Ref Expression
recexprlemex (𝐴P → (𝐴 ·P 𝐵) = 1P)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem recexprlemex
StepHypRef Expression
1 recexpr.1 . . . 4 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
21recexprlemss1l 7633 . . 3 (𝐴P → (1st ‘(𝐴 ·P 𝐵)) ⊆ (1st ‘1P))
31recexprlem1ssl 7631 . . 3 (𝐴P → (1st ‘1P) ⊆ (1st ‘(𝐴 ·P 𝐵)))
42, 3eqssd 3172 . 2 (𝐴P → (1st ‘(𝐴 ·P 𝐵)) = (1st ‘1P))
51recexprlemss1u 7634 . . 3 (𝐴P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P))
61recexprlem1ssu 7632 . . 3 (𝐴P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵)))
75, 6eqssd 3172 . 2 (𝐴P → (2nd ‘(𝐴 ·P 𝐵)) = (2nd ‘1P))
81recexprlempr 7630 . . . 4 (𝐴P𝐵P)
9 mulclpr 7570 . . . 4 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
108, 9mpdan 421 . . 3 (𝐴P → (𝐴 ·P 𝐵) ∈ P)
11 1pr 7552 . . 3 1PP
12 preqlu 7470 . . 3 (((𝐴 ·P 𝐵) ∈ P ∧ 1PP) → ((𝐴 ·P 𝐵) = 1P ↔ ((1st ‘(𝐴 ·P 𝐵)) = (1st ‘1P) ∧ (2nd ‘(𝐴 ·P 𝐵)) = (2nd ‘1P))))
1310, 11, 12sylancl 413 . 2 (𝐴P → ((𝐴 ·P 𝐵) = 1P ↔ ((1st ‘(𝐴 ·P 𝐵)) = (1st ‘1P) ∧ (2nd ‘(𝐴 ·P 𝐵)) = (2nd ‘1P))))
144, 7, 13mpbir2and 944 1 (𝐴P → (𝐴 ·P 𝐵) = 1P)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  {cab 2163  cop 3595   class class class wbr 4003  cfv 5216  (class class class)co 5874  1st c1st 6138  2nd c2nd 6139  *Qcrq 7282   <Q cltq 7283  Pcnp 7289  1Pc1p 7290   ·P cmp 7292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-i1p 7465  df-imp 7467
This theorem is referenced by:  recexpr  7636
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